Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}5x-3y &= -7 \\ 5x+3y &= -4\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $5x = -3y-4$ Divide both sides by $5$ to isolate $x$ $x = {-\dfrac{3}{5}y - \dfrac{4}{5}}$ Substitute this expression for $x$ in the first equation. $5({-\dfrac{3}{5}y - \dfrac{4}{5}}) - 3y = -7$ $-3y - 4 - 3y = -7$ Simplify by combining terms, then solve for $y$ $-6y - 4 = -7$ $-6y = -3$ $y = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $y$ in the top equation. $5x-3( \dfrac{1}{2}) = -7$ $5x-\dfrac{3}{2} = -7$ $5x = -\dfrac{11}{2}$ $x = -\dfrac{11}{10}$ The solution is $\enspace x = -\dfrac{11}{10}, \enspace y = \dfrac{1}{2}$.